An internal or external electric field F can drive the chemical structure, bond order alternation, and electronic structure of linear polymethine dyes from a neutral, bond-alternated, polyene-like structure, through a cyanine-like structure, and ultimately to a zwitterionic (charge-separated) bond-alternated structure. As the structure evolves under the influence of F, the linear polarizability alpha, the first hyperpolarizability beta, and the second hyperpolarizability gamma are seen to be derivatives, with respect to F, of their next lower order polarization (for alpha) or polarizability (for beta and gamma). These derivative relations provide a unified picture of the dependence of the polarizability and hyperpolarizabilities on the structure in linear polymethine dyes. In addition, they allow for predictions of structure-property relations of higher order hyperpolarizabilities.